Let \(f(z):=\mu \exp (z)\)  where \(\mu \in {\mathbb {C}}\)  is chosen
such that \(\bigcup _{n\ge 0} f^n(0)\)  is dense in \({\mathbb {C}}\) .
The existence of such parameters is well-known.
By [1]}, the function \(f\)  has infinitely
many fixed points. Since \({\mathcal {S}}(f)=\lbrace 0\rbrace \) ,
at most one of them is non-repelling
[2]},
so we can pick a repelling fixed point \(z_0\)  of
\(f\) . Let \(L\)  be a linearizer of \(f\)  at \(z_0\) . It follows
from the functional equation that 0 is an omitted
value of \(L\) . By Proposition REF , every
point \(w_n:=f^n(0)\)  is an asymptotic value of \(L\) . It
is also not hard to check that \(L\)  has a direct
singularity lying over each of the points \(w_n\) .
(For a clarification of terminology, see e.g. [3]};
our last claim also follows from [3]}, which is formulated
for linearizers of rational maps only, but extends to
linearizers of transcendental entire maps with the same proof.)
Hence \(L\)  is a map for which the set of projections of direct singularities
(or direct asymptotic values) is dense in \({\mathbb {C}}\) . This is optimal, since
by a theorem of Heins [5]}, the set of
projections of direct singularities is always countable.
